Adiabatic Perturbation Theory in Quantum Dynamics

✍ Scribed by Stefan Teufel

• Publisher: Springer Berlin Heidelberg
• Year: 2008
• Tongue: English
• Leaves: 238
• Series: Lecture Notes in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

Separation of scales plays a fundamental role in the understanding of the dynamical behaviour of complex systems in physics and other natural sciences. A prominent example is the Born-Oppenheimer approximation in molecular dynamics. This book focuses on a recent approach to adiabatic perturbation theory, which emphasizes the role of effective equations of motion and the separation of the adiabatic limit from the semiclassical limit. A detailed introduction gives an overview of the subject and makes the later chapters accessible also to readers less familiar with the material. Although the general mathematical theory based on pseudodifferential calculus is presented in detail, there is an emphasis on concrete and relevant examples from physics. Applications range from molecular dynamics to the dynamics of electrons in a crystal and from the quantum mechanics of partially confined systems to Dirac particles and nonrelativistic QED.

✦ Table of Contents

Title......Page 1
Table of Contents......Page 5
1 Introduction.pdf......Page 7
1.1 The time-adiabatic theorem of quantum mechanics......Page 12
1.2.1 Molecular dynamics......Page 21
1.2.2 The Dirac equation with slowly varying potentials......Page 27
1.3 Outline of contents and some left out topics......Page 33
2.1 The classical time-adiabatic result......Page 38
2.2 Perturbations of fibered Hamiltonians......Page 44
2.3 Time-dependent Born-Oppenheimer theory: Part I......Page 49
2.3.1 A global result......Page 51
2.3.2 Local results and e.ective dynamics......Page 55
2.3.3 The semiclassical limit: .rst remarks......Page 62
2.3.4 Born-Oppenheimer approximation in a magnetic field and Berry’s connection......Page 66
2.4.1 The classical problem......Page 67
2.4.2 A quantum mechanical result......Page 70
2.4.3 Comparison......Page 72
3 Space-adiabatic perturbation theory.pdf......Page 75
3.1 Almost invariant subspaces......Page 79
3.2 Mapping to the reference space......Page 87
3.3 Effective dynamics......Page 93
3.3.1 Expanding the effective Hamiltonian......Page 96
3.4 Semiclassical limit for effective Hamiltonians......Page 99
3.4.1 Semiclassical analysis for matrix-valued symbols......Page 100
3.4.2 Geometrical interpretation: the generalized Berry connection......Page 105
3.4.3 Semiclassical observables and an Egorov theorem......Page 106
4.1 The Dirac equation with slowly varying potentials......Page 109
4.1.1 Decoupling electrons and positrons......Page 110
4.1.2 Semiclassical limit for electrons: the T-BMT equation......Page 115
4.1.3 Back-reaction of spin onto the translational motion......Page 119
4.2 Time-dependent Born-Oppenheimer theory: Part II......Page 128
4.3 The time-adiabatic theorem revisited......Page 131
4.4 How good is the adiabatic approximation?......Page 135
4.5 The Born-Oppenheimer approximation near a conical eigenvalue crossing......Page 140
5 Quantum dynamics in periodic media.pdf......Page 145
5.1 The periodic Hamiltonian......Page 149
5.2 Adiabatic perturbation theory for Bloch bands......Page 155
5.2.1 The almost invariant subspace......Page 159
5.2.2 The intertwining unitaries......Page 163
5.2.3 The effective Hamiltonian......Page 165
5.3 Semiclassical dynamics for Bloch electrons......Page 167
6 Adiabatic decoupling without spectral gap.pdf......Page 176
6.1 Time-adiabatic theory without gap condition......Page 177
6.2 Space-adiabatic theory without gap condition......Page 181
6.3.1 Formulation of the problem......Page 188
6.3.2 Mathematical results......Page 196
A.1 Weyl quantization and symbol classes......Page 205
A.2 Composition of symbols: the Weyl-Moyal product......Page 210
B Operator-valued Weyl calculus for $\tau$ -equivariant symbols......Page 217
C.1 Locally isospectral effective Hamiltonians......Page 223
C.2 Simultaneous adiabatic and semiclassical limit......Page 225
C.3 The work of Blount and of Littlejohn et al.......Page 226
List of Symbols......Page 227
References......Page 229
Index......Page 237